Hard 6×6 cage-sum puzzle #81.

Puzzle signature

Hard profile: 13 cages · 8 distinct cage totals · target range 4–14 · 4 two-cell cages · logic score 131. Use this profile as a quick map of the board's strongest cage constraints before you replay the round.

Puzzle analysis

This PUZRU archive puzzle uses 13 connected sum cages across a 6×6 grid with 6 carried fixed values. Its cage profile contains 4 two-cell, 8 three-cell and 1 four-cell cages. Target sums range from 4 to 14, with an average cage target of 9.7.

The generator assigned a logic score of 131, which PUZRU classifies as Hard. The archived quality record verifies one exact solution and a logical solve path without random guessing.

What makes this round distinct

This board combines 8 distinct cage totals. It contains 1 low-target cage (5 or below) and 0 high-target cages (18 or above), creating a round-specific balance between constrained low combinations and high-digit pressure. Its 4 two-cell cages create especially short candidate lists that interact tightly with row and column exclusions.

Logical deduction trace

1. R1C3 starts fixed at 6 from the Chain carry.
2. R2C5 starts fixed at 6 from the Chain carry.
3. R3C4 starts fixed at 4 from the Chain carry.
4. R4C6 starts fixed at 2 from the Chain carry.
5. R5C2 starts fixed at 4 from the Chain carry.
6. R6C1 starts fixed at 5 from the Chain carry.
7. Cage R4C2, R5C2 totals 10; this removes 1/3/5 from R4C2.
8. Cage R3C6, R4C6 totals 8; this removes 1/3/5 from R3C6.
9. Cage R5C3, R6C2, R6C3 totals 6; this removes 5 from R5C3.
10. Cage R5C3, R6C2, R6C3 totals 6; this removes 6 from R6C2.
11. Cage R3C4, R4C3, R4C4 totals 10; this removes 3/4 from R4C3.
12. Cage R3C4, R4C3, R4C4 totals 10; this removes 3/6 from R4C4.
13. Cage R2C5, R3C5 totals 7; this removes 2/3/5 from R3C5.
14. Cage R1C4, R2C4 totals 4; this removes 2/5 from R1C4.
15. Cage R1C4, R2C4 totals 4; this removes 2/5 from R2C4.
16. Cage R4C1, R5C1, R6C1 totals 14; this removes 1/4 from R4C1.
17. Cage R4C1, R5C1, R6C1 totals 14; this removes 1/2 from R5C1.
18. Cage R1C3, R2C2, R2C3 totals 12; this removes 3 from R2C2.
19. Cage R1C3, R2C2, R2C3 totals 12; this removes 2/3 from R2C3.
20. Cage R5C4, R6C4, R6C5 totals 11; this removes 1 from R6C5.
21. Cage R1C1, R1C2, R2C1 totals 10; this removes 1/2 from R1C2.
22. R3C5 has only candidate 1 after row, column and cage elimination. Set R3C5 = 1.
23. R3C6 has only candidate 6 after row, column and cage elimination. Set R3C6 = 6.
24. R4C2 has only candidate 6 after row, column and cage elimination. Set R4C2 = 6.

How to approach this 6×6 puzzle

• Start with the two-cell cages: their shorter combination lists make them efficient places to eliminate candidates against row and column restrictions.
• Low-total cages are equally useful: only a small set of digit combinations can reach those sums without repeats elsewhere in the grid.
• After every cage deduction, scan both the row and the column before committing a value. PUZRU puzzles are designed so cage arithmetic and Latin-square constraints reinforce each other.

The live result

GGOGUZ became the Discoverer by submitting the first verified solution accepted by the global authority. After that moment, the shared 45-second contribution window remained open and the puzzle finished with 1 verified solver.

Six values continued into the Chain

The PUZRU world never resets to a completely unrelated board. Six solved positions from this puzzle were carried into the next puzzle as fixed values. For Puzzle #81, the carried values were 3 2 1 5 6 2.

This carry rule makes every puzzle both a standalone logic challenge and one link in an ongoing global sequence.

Chain proof

The public archive records the completed puzzle and its cryptographic Chain hash so later rounds can be linked back to the same history.

ixi99cuNoK2xHfTDW9m5a5ZTQfuDR0bcOt0l14NzT1g

Archive replay is unranked: it never changes the live Chain or leaderboard.

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